Tag Archives: modules

Idempotents and Decomposition

Let R be a general ring, not necessarily commutative. An element x∈R is said to be idempotent if x2 = x. Note An endomorphism f of an R-module M (i.e. ) is an idempotent if and only if f is a projection, i.e. M = ker(f) ⊕ im(f) and f … Continue reading

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Tensor Product over Noncommutative Rings

Following the earlier article on tensor products of vector spaces, we will now look at tensor products of modules over a ring R, not necessarily commutative. It turns out we have to distinguish between left and right modules now. Indeed recall … Continue reading

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Hom Functor

Fret not if you’re unfamiliar with the term functor; it’s a concept in category theory we will use implicitly without delving into the specific definition. This topic is, unfortunately, a little on the dry side but it’s a necessary evil to get … Continue reading

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Exact Sequences and the Grothendieck Group

As before, all rings are not commutative in general. Definition. An exact sequence of R-modules is a collection of R-modules and a sequence of R-module homomorphisms: such that for all i. Examples 1. The sequence is exact if and only if f … Continue reading

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Composition Series

Positive integers can be uniquely factored as a product of primes. Here, we would like to prove a counterpart for modules. Now there are two ways to “factor” a module M; a more liberal way takes a submodule N which gives us composition … Continue reading

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Radical of Module

As mentioned in the previous article, we will now describe the “bad elements” in a ring R which stops it from being semisimple. Consider the following ring: Since R is finite-dimensional over the reals, it is both artinian and noetherian. However, R is not … Continue reading

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The Group Algebra (I)

[ Note: the contents of this article overlap with a previous series on character theory. ] Let K be a field and G a finite group. The group algebra K[G] is defined to be a vector space over K with basis , where “g” here is … Continue reading

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